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GATE 2025 PH – Question 22

Solid State Physics · free electron theory · 1 mark · Multiple choice

Consider one mole of a monovalent metal at absolute zero temperature, obeying the free electron model. Its Fermi energy is $E_F$. The energy corresponding to the filling of $N_A/2$ electrons, where $N_A$ is Avogadro number, is $2^nE_F$. The value of $n$ is

  1. $-2/3$
  2. $2/3$
  3. $-1/3$
  4. $-1$

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Correct answer: (A) $-2/3$

Explanation

For free electrons in three dimensions, the number of filled states up to energy $E$ is
$$N(E)\propto E^{3/2}.$$

At absolute zero all states up to the Fermi energy $E_F$ are filled by the $N_A$ electrons:
$$N_A\propto E_F^{3/2}.$$

**For $N_A/2$ electrons:**
$$\frac{N_A}{2}\propto E^{3/2}\;\Rightarrow\;\left(\frac E{E_F}\right)^{3/2}=\frac12\;\Rightarrow\;E=2^{-2/3}E_F .$$

So $2^nE_F=2^{-2/3}E_F$ and $n=\mathbf{-\tfrac23}$ (option A).